It is useful to simply investigate **angular momentum with just one free rotation angle**.
This might be the quantum plane propeller.
We will do a good job of this in 3 dimensions later.

Lets assume we have a **mass constrained to move in a circle of radius **.
Assume the motion in the circle is free, so there is no potential.
The kinetic energy is
for
.

If we measure distance around the circle, then
and the one problem we have is that once I go
completely around the circle, I am back to
.
Lets just go ahead and write our wavefunction.

Remembering angular momentum, lets call the combination . Our wave is .

This must be **single valued** so we need to require that

The

This will prove to be true for 3 dimensions too, however, the 3 components of angular momentum do not commute with each other, leading to all kinds of fun.

Jim Branson 2013-04-22